Here are 10 similar problems based on the quadratic equation $y = -x^2 + 2x + 3$, focusing on determining the characteristics of quadratic functions:

1. $y = x^2 - 4x + 5$
2. $y = -2x^2 + 3x - 1$
3. $y = x^2 + 6x + 9$
4. $y = -x^2 - 5x + 4$
5. $y = 3x^2 - 2x + 7$
6. $y = -4x^2 + 8x + 1$
7. $y = x^2 + 2x - 3$
8. $y = -x^2 + 4x - 6$
9. $y = 2x^2 - 5x + 3$
10. $y = -3x^2 + 6x - 9$

These problems ask you to determine the vertex, domain, range, x- and y-intercepts, and whether each equation represents a function. If you want more problems like these or would like detailed explanations on any of these, feel free to ask!

Would you like any clarifications or details on these problems?

Here are 5 related questions for further exploration:

1. How do you find the vertex of a quadratic equation?
2. What is the domain and range of a standard quadratic function?
3. How can you determine the x- and y-intercepts of a quadratic function?
4. Why is every quadratic equation considered a function?
5. What determines whether a parabola opens upwards or downwards?

Tip: The coefficient of $x^2$ determines whether a parabola opens upwards (positive coefficient) or downwards (negative coefficient).